Trigonometry Examples

$\cos^{6} (a) + \sin^{6} (a)$

Step 1

Apply basic rules of exponents.

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Step 1.1

Rewrite $\cos^{6} (a)$ as ${(\cos^{2} (a))}^{3}$ .

${(\cos^{2} (a))}^{3} + \sin^{6} (a)$

Step 1.2

Rewrite $\sin^{6} (a)$ as ${(\sin^{2} (a))}^{3}$ .

${(\cos^{2} (a))}^{3} + {(\sin^{2} (a))}^{3}$

Step 2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = \cos^{2} (a)$ and $b = \sin^{2} (a)$ .

$(\cos^{2} (a) + \sin^{2} (a)) ({(\cos^{2} (a))}^{2} - \cos^{2} (a) \sin^{2} (a) + {(\sin^{2} (a))}^{2})$

Step 3

Rearrange terms.

$(\sin^{2} (a) + \cos^{2} (a)) ({(\cos^{2} (a))}^{2} - \cos^{2} (a) \sin^{2} (a) + {(\sin^{2} (a))}^{2})$

Step 4

Apply pythagorean identity.

$1 ({(\cos^{2} (a))}^{2} - \cos^{2} (a) \sin^{2} (a) + {(\sin^{2} (a))}^{2})$

Step 5

Simplify terms.

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Step 5.1

Multiply ${(\cos^{2} (a))}^{2} - \cos^{2} (a) \sin^{2} (a) + {(\sin^{2} (a))}^{2}$ by $1$ .

${(\cos^{2} (a))}^{2} - \cos^{2} (a) \sin^{2} (a) + {(\sin^{2} (a))}^{2}$

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply the exponents in ${(\cos^{2} (a))}^{2}$ .

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Step 5.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${\cos (a)}^{2 \cdot 2} - \cos^{2} (a) \sin^{2} (a) + {(\sin^{2} (a))}^{2}$

Step 5.2.1.2

Multiply $2$ by $2$ .

$\cos^{4} (a) - \cos^{2} (a) \sin^{2} (a) + {(\sin^{2} (a))}^{2}$

Step 5.2.2

Multiply the exponents in ${(\sin^{2} (a))}^{2}$ .

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Step 5.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos^{4} (a) - \cos^{2} (a) \sin^{2} (a) + {\sin (a)}^{2 \cdot 2}$

Step 5.2.2.2

Multiply $2$ by $2$ .

$\cos^{4} (a) - \cos^{2} (a) \sin^{2} (a) + \sin^{4} (a)$